Yes, quoted terms are not necessarily exact substitutions but should be similar in meaning. "It's a fact" vs. "I've got proof," as in many arguments, seems a question of degree. I guess I don't see them as similar. "I've got proof" can be used instead of "It's a fact," but so could "You don't know what you're talking about." "...both things that someone trying to prove a point might say," but they're really quite different.
I don't think you're wrong, but really, do you really want to disagree with Mr. Know-all? (By the way, "Mr. Know-All" by W. Somerset Maugham is one of my favorite short stories. It's about an insufferable bore named Max Kelada, a character with a quality I very much admire --- and so will you if you read carefully.)